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Journal of Convex Analysis 14 (2007), No. 2, 395--412
Copyright Heldermann Verlag 2007

Local Uniform Rotundity in Calderón-Lozanovskii Spaces

Pawel Foralewski
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznan, Poland

Pawel Kolwicz
Institute of Mathematics, Faculty of Electricity, University of Technology, Piotrowo 3a, 60-965 Poznan, Poland

We find criteria for local uniform rotundity of Calderon-Lozanovskii spaces solving problem XII from S. T. Chen, Y. A. Cui, H. Hudzik and T. F. Wang ["On some solved and unsolved problems in geometry of certain classes of Banach function spaces", in: Unsolved Problems on Mathematics for the 21st Century, J. M. Abe & Tanaka (eds.), IOS Press (2001)] and generalizing several theorems, which give only the sufficient (or necessity) conditions [see H. Hudzik, A. Kaminska and M. Mastylo, "Monotonicity and rotundity properties in Banach lattices", Rocky Mountain J. Math. 30(3) (2000) 933--949; cf. also J. Cerda, H. Hudzik and M. Mastylo, "On the geometry of some Calderon-Lozanovskii interpolation spaces", Indagationes Math. N.S. 6(1) (1995) 35-49]. In particular we obtain the respective criteria for Orlicz-Lorentz spaces which has been proved directly in papers of H. Hudzik, A. Kaminska and M. Mastylo ["On geometric properties of Orlicz-Lorentz spaces", Canadian Math. Bull. 40(3) (1997) 316-329] and J. Cerda, H. Hudzik, A. Kaminska and M. Mastylo ["Geometric properties of symmetric spaces with applications to Orlicz-Lorentz spaces", Positivity 2 (1998) 311-337].

Keywords: Koethe space, Calderon-Lozanovskii space , Orlicz-Lorentz space, local uniform rotundity, monotonicity properties.

MSC: 46B20, 46E30, 46B04, 46B42, 46A45

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